Hyperbolic Sine

Author

Eric W. Weisstein
August 24, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/HyperbolicSine.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HyperbolicSine.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plot

<<Utilities`Plot`
<<Utilities`Typesetting`

Plot[Sinh[x],{x,-2.5,2.5},AxesLabel->TraditionalForm/@{x,Sinh[x]},
PlotRange->{Automatic,{-5,5}},PlotStyle->Red]

Complex Plane

ComplexPlot3D[Sinh[z],{z,-5,5}]//Timing

ComplexContourPlot[Sinh[z],{z,-5,5}]//Timing

Special Values

Sinh[0]

Limit[Sinh[x],x->#]&/@{-∞,∞}

Sinh[Log[GoldenRatio]]

FullSimplify[Sinh[Log[GoldenRatio]]]

Series

Series[Sinh[x],{x,0,10}]

V5

<<DiscreteMath`RSolve`

SeriesTerm[Sinh[z],{z,0,n}]

Table[%,{n,0,10}]

V7

SeriesCoefficient[Sinh[x],{x,0,n}]//InputForm

GeneratingFunction[%,n,z]//InputForm

Derivative and Integral

Derivative

D[Sinh[z],z]

Integral

Integrate[Sinh[z],z]

Sum

(∑_n^∞)(x^n) Piecewise[(({{(1/(n!)), n mod 2==1∧n≥0}})),0]==sinh(x)

Sum[Piecewise[{{n!^(-1),Mod[n,2]==1&&n≥0}},0]x^n,{n,∞}]//InputForm

Sum[Piecewise[{{n!^(-1),Mod[n,2]==1&&n≥0}},0]x^n,{n,0,20}]/.x->1.3

Sinh[1.3]

(∑_(n=0)^∞)((z^(2 n+1))/((2 n+1)!))==sinh(z)

Sum[((z^(2n+1))/((2n+1)!)),{n,0,∞}]